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A note on Wick products and the fractional Black-Scholes model
Stockholm School of Economics.
Dept. of Appl. Math. and Statistics, Universitetsparken 5, 2100 Copenhagen, Denmark. (Mathematical Statistics)ORCID iD: 0000-0001-9210-121X
2005 (English)In: Finance and Stochastics, ISSN 0949-2984, E-ISSN 1432-1122, Vol. 9, no 2, 197-209 p.Article in journal (Refereed) Published
Abstract [en]

In some recent papers (Elliott and van der Hoek 2003; Hu and Oksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Oksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Ito integral is replaced by the Wick integral, and proofs have been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The objective of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek (2003), a naive buy-and-hold strategy does not in general qualify as "self-financing". We also show that in Hu and Oksendal (2003), a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative "value".

Place, publisher, year, edition, pages
2005. Vol. 9, no 2, 197-209 p.
Keyword [en]
mathematical finance, fractional Brownian motion, arbitrage
National Category
Probability Theory and Statistics
URN: urn:nbn:se:kth:diva-81931DOI: 10.1007/s00780-004-0144-5ISI: 000228942500003OAI: diva2:497771
QC 20120313Available from: 2012-02-11 Created: 2012-02-11 Last updated: 2012-03-13Bibliographically approved

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Björk, TomasHult, Henrik
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